Weak scalar curvature lower bounds along Ricci flow

Abstract

In this paper, we study Ricci flow on compact manifolds with a continuous initial metric. It was known from Simon that the Ricci flow exists for a short time. We prove that the scalar curvature lower bound is preserved along the Ricci flow if the initial metric has a scalar curvature lower bound in distributional sense provided that the initial metric is W 1,p for some n < p ≤ ∞. As an application, we use this result to study the relation between Yamabe invariant and Ricci flat metrics. We prove that if the Yamabe invariant is nonpositive and the scalar curvature is nonnegative in distributional sense, then the manifold is isometric to a Ricci flat manifold.

Rigidity of Lipschitz map using harmonic map heat flow

“…The idea is similar to that in [22]. Since g 0 ∈ W 1,p for p > n, the works in [18,26] infers that the constructed G-flow g(t) satisfies a better estimate on M ×(0, T ]: Moreover, it is known that each ǧ(t) = λ −1 (t)g(t), t ∈ (0, T ] is isometric to the standard sphere. We now construct the diffeomorphism to compensate the singularity of (4.10) as t → 0.…”

“…With this identification, the metric g is Lipschitz on M and [21, Proposition 5.1] applies to show that R(g) ≥ n(n−1) in the distribution sense. And hence, g satisfies the assumptions of Theorem 1.2 by [18,Corollary 1.2] with f chosen to be identity. Hence, g = h and so does g 0 on Ω.…”

“…Suppose g is a C 0 metric. If g is satisfies one of the following, then it is known that it also has scalar curvature bounded below by σ in the sense of Definition 1.2: (a) g has scalar curvature bounded below by σ in the sense of Burkhardt-Guim [3] using regularizing Ricci flow; (b) g ∈ W 1,p , p > n with R ≥ σ in the sense of distribution as in [21] by Jiang-Sheng-Zhang [18] ; (c) g is smooth away from singularity Σ of co-dimension at least three and has R(g) ≥ σ outside Σ by the work [22] of the authors; (d) there exist smooth g i → g in C 0 norm so that some integral form of lower bound of R(g i ) is satisfied by Huang and the first named author [12].…”

Scalar curvature lower bound under integral convergence

“…The preservation of scalar curvature's lower bound had played a important role in this works. More recently in [6], Jiang, Sheng and Zhang consider the scalar curvature lower bounds in distributional sense for singular metrics in W 1,p for some n < p ≤ ∞. In particular, they prove that the weak notion of scalar curvature lower bound is preserved along Ricci flow.…”

“…Motivated by the above mentioned works, it is natural to ask if the pointwise scalar curvature lower bound can be weakened further to integral forms under C 0 convergence in contrast with the work in [6]. To this end, in this work we generalize the Theorem in two directions.…”